Think you can just make the new Switch your primary console.
If you have a Nintendo Switch Online account most saves should be cloud backed up although I think a few like animal crossing and Pokémon don't support it, not sure in those circumstances with regards save transferring.
accounts.nintendo.com
Go the Shop Menu
Scroll to bottom
You can only do it remotely like this once a year
Whichever console you next use to access Nintendo eShop will automatically be set as the primary console for your Nintendo Account.